PredictabilityThreshold
plain-language theorem explainer
The definition assigns PredictabilityThreshold the value of the J-cost function evaluated at the golden ratio. Climate modelers and Recognition Science researchers cite it to locate the structural horizon where forecast skill on uncertainty ratio r is lost once J(r) meets or exceeds J(φ). The declaration is a direct one-line assignment from the Cost library.
Claim. The predictability-horizon threshold is the real number $J(φ)$, where $J$ is the J-cost function on positive ratios and $φ$ is the golden ratio.
background
The module treats climate forecast skill decay as a recognition event on the initial-condition uncertainty ratio $r := σ_forecast / σ_initial$. The J-cost is the derived cost of a multiplicative recognizer comparator, given explicitly by $J(x) = (x + x^{-1})/2 - 1$. Upstream, Cost.Jcost is the same function used for recognition events in ObserverForcing and MultiplicativeRecognizerL4. The module doc states that deterministic skill is permitted while $J(r) < J(φ)$ and lost once $J(r) ≥ J(φ)$, placing the horizon at the canonical golden-section quantum.
proof idea
Direct definition that binds PredictabilityThreshold to the term Cost.Jcost phi. No tactics or lemmas are applied; the body is a single equality to the imported Jcost evaluator.
why it matters
This supplies the fixed threshold referenced by IsPastHorizon, IsWithinHorizon, and ClimatePredictabilityCert. It instantiates the module claim that the same J(φ) band (0.11, 0.13) gates climate predictability as it does plaque vulnerability, magnetic reconnection, and other recognition thresholds. It sits inside the T5 J-uniqueness step of the forcing chain and closes the structural prediction for chaotic systems.
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